Section 12–2 Friction

(Drag force, F ≈ cv² / Static friction, F = μN / Sliding friction, F = μN)

In this section, Feynman discusses drag forces, static friction, and sliding friction (or kinetic friction).

1. Drag force, F ≈ cv²:

“…it is a remarkable fact that the drag force on an airplane is approximately a constant times the square of the velocity, or F ≈ cv² (Feynman et al., 1963, section 12–2 Friction).”

The drag force on an airplane is approximately a constant times the square of the velocity, F ≈ cv². According to Feynman, if the velocity of an airplane is extremely low, then the law of drag force needs to be modified because the drag force is directly proportional to the velocity (or approximately linear dependence). He explains that the drag force on an airplane flying through the air is not a simple law as it involves air molecules rushing over the wings, the swirling in the back, the changes going on around the fuselage, and many complications. Interestingly, it is possible for a drag force (or wave drag) to be proportional to the cube of the velocity, F ≈ kv³. Wave drag may occur as a result of pressure differences around a moving object in between two fluids such as air and water (Blazevich, 2017).

Feynman says that the drag force expressed in terms of F = cv² begin to fail at higher speeds not simply due to slight changes in the coefficient. He elaborates that the force on a wing depends upon the other wing and the rest of the plane. As an alternative, the drag force can be represented by the equation F = ½CrAv² where C is the drag coefficient, r is the density of the fluid, and A is the cross-sectional area of the object facing the fluid. Similarly, Feynman uses the formula C_D = F/[½rv²dl] for the drag force on a circular cylinder (Feynman et al, 1964, section 41–4). To be more precise, the drag force depends on the fluid’s viscosity and compressibility, the shape of the body, and the body’s inclination to the flow. The drag force equation can be derived to within a multiplicative constant by using dimensional analysis.

2. Static Friction, F = μN:

“… the frictional force is proportional to this normal force, and has a more or less constant coefficient; that is, F = μN, where μ is called the coefficient of friction (Feynman et al., 1963, section 12–2 Friction).”

Feynman distinguishes two kinds of drag forces that are due to fast movement in the air and slow movement in honey. He states that there is another kind of friction (dry friction or sliding friction) which occurs when one solid body slides on another. One may expect Feynman to explain that static friction is not really static because there are micro-displacements of atoms. More important, the formula F = μ_sN is approximately correct for static friction and it can be demonstrated by a simple experiment: we place a block of weight W on an inclined plane and measure the angle θ which the block begins to slide. This occurs when the component of the weight parallel to the plane (Wsin θ) is equal to the maximum static frictional force μ_sN (or μWcos θ). By equating the two expressions, we can deduce the static coefficient of friction, μ_s = tan θ.

The empirical law F = μ_sN has its limitations, such that it does not always work. Firstly, we should notice that when the inclined plane is tilted at the correct angle θ, the block does not slide steadily but in a halting fashion. The variations of coefficient μ_s are caused by different degrees of smoothness or hardness of the material, and possibly dirt, impurities, or oxides. Historically, there are two laws of static friction: (1) Static frictional force (μ_sN) is proportional to the normal reaction, N (Amontons’ first law). (2) Static frictional force is independent of the apparent area of contact, A (Amontons’ second law). However, the frictional force can be represented by the equation F = μN + kA, where kA is dependent on the area of contact between two surfaces.

3. Sliding friction, F = μN:

“…the friction to be overcome to get something started (static friction) exceeds the force required to keep it sliding (sliding friction), but with dry metals, it is very hard to show any difference (Feynman et al., 1963, section 12–2 Friction).”

Coulomb’s law of friction can be described as follows: the kinetic frictional force is independent of sliding speed. Feynman elaborates that the frictional force is not well understood and it is difficult to perform accurate experiments in friction. The apparent decreases of the kinetic frictional force at high speeds are often due to vibrations. Currently, physicists may explain that the kinetic frictional force decreases for materials such as steel, copper, and lead, but increases for the polymer Teflon (Besson et al., 2007). In general, frictional forces are influenced by many factors such as surface cleanliness, surface roughness, contact temperature, relative humidity, lubricant properties and presence of loose particles (Blau, 2008).

Many simply believe that the static friction exceeds the kinetic friction, but it is hard to tell any difference for dry metals. Feynman explains that the opinion may arise from experiences where some oils or lubricants are present, without using the term wet friction. If we try to have pure copper by cleaning and take every conceivable precaution, it is still difficult to determine the coefficient μ. It is even possible that two pieces of copper stick together if we tilt the apparatus to a vertical position. One reason for this behavior is that when the atoms in contact are of the same kind, there is no way for the atoms to “know” that they are in different pieces of copper. Physics teachers may conclude the section using the words of American Society of Mechanics Handbook (1992): “[u]niversal agreement as to what truly causes friction does not exist and much still remains to be done before a complete picture can emerge (p. 27).”

Questions for discussion:

1. Could a drag force be directly proportional to the velocity of an object as well as the square of the velocity of the same object simultaneously?

2. What are the idealizations, approximations, and limitations in conceptualizing static friction?

3. What are the idealizations, approximations, and limitations in conceptualizing kinetic friction?

The moral of the lesson: frictional forces (approximately μN) are due to molecular forces that cannot be satisfactorily explained by classical physics; we need quantum mechanics to understand them fully.

References:

1. American Society of Mechanics (1992). Friction, Lubrication, and Wear Technology. ASM Handbook, Vol. 18. Ohio: ASM International.

2. Besson, U., Borghi, L., De Ambrosis, A., & Mascheretti, P. (2007). How to teach friction: Experiments and models. American Journal of Physics, 75(12), 1106-1113.

3. Blau, P. J. (2008). Friction science and technology: from concepts to applications (2nd ed.). Boca Raton: CRC press.

4. Blazevich, A. J. (2017). Sports Biomechanics: The Basics: Optimising Human Performance. London: A & C Black.

5. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.